Cycle Isolation of Graphs with Small Girth

Let G be a graph. A subset \(D \subseteq V(G)\) is a decycling set of G if \(G-D\) contains no cycle. A subset \(D \subseteq V(G)\) is a cycle isolating set of G if \(G-N[D]\) contains no cycle. The decycling number and cycle isolation number of G, denoted by \(\phi (G)\) and \(\iota _c(G)\), are the minimum cardinalities of a decycling set and a cycle isolating set of G, respectively. Dross, Montassier and Pinlou (Discrete Appl Math 214:99–107, 2016) conjectured that if G is a planar graph of size m and girth at least g, then \(\phi (G) \le \frac{m}{g}\). So far, this conjecture remains open. Recently, the authors proposed an analogous conjecture that if G is a connected graph of size m and girth at least g that is different from \(C_g\), then \(\iota _c(G) \le \frac{m+1}{g+2}\), and they presented a proof for the initial case \(g=3\). In this paper, we further prove that for the cases of girth at least 4, 5 and 6, this conjecture is true. The extremal graphs of results above are characterized.